Measure of association between non-dichotomous ordinal variable and continuous variable

Question

I have to collect and then analyse a large dataset. A particular data input (continuous variable) is laborious to collect and I believe that an ordinal category (1,2,3,4) would correlate closely with this continuous variable. I want to test this after collecting 10% of the data including the continuous variable, and provided there is a correlation, proceed to collect the other 90% of the data without collecting the continuous variable (simplifying my data collection greatly). At this stage I am not sure if the data will be normally distributed.

What is the appropriate statistical test to do this properly?

Answer 1

An appropriate test would be a rank correlation, like Spearman's rho or Kendall's tau. For instance, using R:

> ordinals <- rep(1:4,each=100)
> set.seed(1)
> continuous <- rnorm(length(ordinals),ordinals,1)
> cor.test(continuous,ordinals,method="kendall")

        Kendall's rank correlation tau

data:  continuous and ordinals
z = 16.3381, p-value < 2.2e-16
alternative hypothesis: true tau is not equal to 0
sample estimates:
     tau 
0.610561 

Nevertheless, I'd always recommend looking at plots of your data to get a visual impression of whether the relationship is "good enough". Scatterplots or beeswarm plots are better for this than boxplots, which lose too much information:

> require(beeswarm)
> beeswarm(continuous~ordinals,pch=19)

Answer 2

I suggest Somers' $D_{xy}$ or $D_{yx}$ depending on which variable is $x$ and which is $y$. This will result in a concordance-based measure that does not penalize for ties in the 4-level variable and does penalize for any ties in the continuous variable. In R's Hmisc package the rcorr.cens function is one of probably many ways to get this, along with a standard error. If you want to penalize for ties in the 4-level ordinal variable then one of Kendall's $\tau$s or Spearman's $\rho$ may be appropriate.